How do I move an object with a curve modifier applied to it?
If you select both the claw and the deforming curve and press Alt+D to do a linked duplicate of both of them, wouldn't that fix it? 

You can then move the claw to a new position by selecting the duplicate claw and the duplicate curve, doing any transformation you want to both of them.

Because you did a 'linked duplicate' you can then still edit any of the curve objects and they will all update.

Here I have duplicated the curve and claw several times (using 'linked duplicate'), scaled them, moved them and by going into edit mode of a single curve I can edit them all at once: